Laminar flow through isotropic granular porous media

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(1)Laminar flow through Isotropic Granular Porous Media. Sonia Woudberg. Thesis presented in partial fulfilment of the requirements for the degree of Master of Engineering Science at the University of Stellenbosch.. Promoter: Prof. J.P. du Plessis. December 2006.

(2) Declaration. I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.. Date:. Signature:. i.

(3) Abstract. An analytical modelling procedure for predicting the streamwise pressure gradient for steady laminar incompressible flow of a Newtonian fluid through homogeneous isotropic granular porous media is introduced. The modelling strategy involves the spatial volume averaging of a statistical representative portion of the porous domain to obtain measurable macroscopic quantities from which macroscopic transport equations can be derived. A simple pore-scale model is introduced to approximate the actual complex granular porous microstructure through rectangular cubic geometry. The sound physical principles on which the modelling procedure is based avoid the need for redundant empirical coefficients. The model is generalized to predict the rheological flow behaviour of non-Newtonian purely viscous power law fluids by introducing the dependence of the apparent viscosity on the shear rate through the wall shear stress. The field of application of the Newtonian model is extended to predict the flow behaviour in fluidized beds by adjusting the Darcy velocity to incorporate the relative velocity of the solid phase. The Newtonian model is furthermore adjusted to predict fluid flow through Fontainebleau sandstone by taking into account the effect of blocked throats at very low porosities. The analytical model as well as the model generalizations for extended applicability is verified through comparison with other analytical and semi-empirical models and a wide range of experimental data from the literature. The accuracy of the predictive analytical model reveals to be highly acceptable for most engineering designs.. ii.

(4) Opsomming. ’n Analitiese modelleringsprosedure is bekend gestel om die stroomsgewyse drukgradiënt vir tydonafhanklike, laminêre, onsaamdrukbare vloei van ’n Newtoniese vloeistof deur homogene, isotrope poreuse media met ’n korrelstruktuur te voorspel. Die modelleringstrategie berus op die ruimtelike volumetriese gemiddelde van ’n statisties-verteenwoordigende gedeelte van die poreuse medium om meetbare makroskopiese groothede te verkry waarvan makroskopiese oordragvergelykings afgelei kan word. ’n Eenvoudige porie-skaal model word voorgestel om die werklike komplekse korrelagtige mikro-struktuur deur ’n reghoekige kubiese geometrie te benader. Die fisiese grondbeginsels waarop die modelleringstrategie gegrond is, vermy die behoefte vir empiriese koëffisiënte. Die model is veralgemeen om die reologiese vloeigedrag van nie-Newtoniese, suiwer viskeuse, magswet-vloeistowwe te voorspel deur die afhanklikheid van die effektiewe viskositeit op die skuifspanningstempo in te voer deur die skuifspanning op die wand. The toepassingsveld van die Newtoniese model is uitgebrei om die vloeigedrag in sweefbeddens te voorspel deur die Darcy snelheid aan te pas om sodoende die relatiewe snelheid van die vastestoffase in berekening te bring. Die Newtoniese model is verder aangepas om die vloei van vloeistowwe deur Fontainebleau sandsteen te voorspel deur die effek van geblokkeerde kanale by baie lae porositeite in ag te neem. Die analitiese model, sowel as die veralgemenings van die model vir uitgebreide toepasbaarheid, is geverifieer deur vergelyking met ander analitiese en semi-empiriese modelle en ’n wye verskeidenheid eksperimentele data vanuit die literatuur. Die akkuraatheid van die voorspelbare analitiese model blyk hoogs aanvaarbaar te wees vir die meeste ingenieursontwerpe.. iii.

(5) Acknowledgements. I wish to express my sincere gratitude to the following people who contributed to this study by inspiring me in their own special way: • God, for guiding me in life and giving me the potential to follow this path. • My supervisor, Prof. Prieur du Plessis, for not only guiding me to face the challenges of our competitive world, but also showing me the world and to appreciate the priceless things in life. • My parents, Johann and Linda Woudberg, for the best moral and financial support one could wish for. • My sister and bother-in-law, Tania and André Heunis, for their concern and encouragement. • My family and friends for their wishes of support when I was overseas. • Our head of division, Dr. Francois Smit, for his moral support at times when it mattered the most and the financial support from BIWUS to attend the conference in India. • Prof. Mark Knacksteadt for hosting me in Australia, Prof. Britt Halvorsen for hosting me in Norway and Prof. Jack Legrand for financial assistance to visit France. • The South African National Research Foundation (NRF) for the Prestigious Scholarship and the additional Travel Grant.. iv.

(6) Contents. 1 Introduction. 1. 1.1. Granular models from literature . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.3. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 1.4. Layout of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2 Method of Volume Averaging. 6. 2.1. Interstitial transport equations . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2. Representative Elementary Volume . . . . . . . . . . . . . . . . . . . . . .. 7. 2.3. Macroscopic volume averaged quantities . . . . . . . . . . . . . . . . . . .. 8. 2.4. Macroscopic transport equations . . . . . . . . . . . . . . . . . . . . . . . .. 9. 3 Rectangular Granular Pore-Scale Model. 10. 3.1. Rectangular Representative Unit Cell . . . . . . . . . . . . . . . . . . . . . 10. 3.2. Staggered and non-staggered arrays . . . . . . . . . . . . . . . . . . . . . . 11 3.2.1. Fully staggered array . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 3.2.2. Regular array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 3.3. Piece-wise straight streamlines . . . . . . . . . . . . . . . . . . . . . . . . . 15. 3.4. Volume partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. 3.5. Volume averaging of transport equations over an RUC . . . . . . . . . . . 18. v.

(7) 3.6. Classification of laminar flow regimes . . . . . . . . . . . . . . . . . . . . . 20 3.6.1. Limit of low Reynolds number flow . . . . . . . . . . . . . . . . . . 20. 3.6.2. Transition regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 3.6.3. Steady laminar limit of the inertial flow regime . . . . . . . . . . . 21. 4 Low Reynolds Number Flow Regime 4.1. 4.2. 23. Closure modelling at low to moderate porosities . . . . . . . . . . . . . . . 23 4.1.1. Evaluation of the coefficient β . . . . . . . . . . . . . . . . . . . . . 30. 4.1.2. Isotropic RUC model . . . . . . . . . . . . . . . . . . . . . . . . . . 32. Comparison with granular models from literature . . . . . . . . . . . . . . 33 4.2.1. Hydraulic diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 4.2.2. Dimensionless permeability and shear factor . . . . . . . . . . . . . 35. 4.2.3. Coefficient A in the Blake-Kozeny equation . . . . . . . . . . . . . . 38. 4.2.4. High porosity model . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 4.2.5. Asymptote matching of low and high porosity models . . . . . . . . 41. 4.2.6. The Kozeny constant . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 5 Laminar inertial flow regime. 46. 5.1. Closure modelling at moderate to high porosities. . . . . . . . . . . . . . . 46. 5.2. Comparison with granular models from literature . . . . . . . . . . . . . . 52 5.2.1. Shear factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. 5.2.2. Coefficient B in the Burke-Plummer equation . . . . . . . . . . . . 52. 6 Asymptote matching of laminar limits. 55. 6.1. Comparison with the Ergun equation . . . . . . . . . . . . . . . . . . . . . 56. 6.2. Critical Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . 58. 7 Non-Newtonian flow. 59 vi.

(8) 7.1. RUC model for purely viscous power law flow . . . . . . . . . . . . . . . . 61 7.1.1. 7.2. Comparison with empirical models from literature . . . . . . . . . . 63. Asymptote matching of the shear stress . . . . . . . . . . . . . . . . . . . . 71. 8 Model Applications 8.1. Fluidized Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.1.1. 8.2. 80. Comparison of different drag models . . . . . . . . . . . . . . . . . 83. Sandstones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87. 9 Conclusions and Recommendations. 91. A Volume averaging of transport equations. 94. A.1 Volume averaging theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A.2 Volume averaging of the continuity equation . . . . . . . . . . . . . . . . . 95 A.3 Volume averaging of the Navier-Stokes equation . . . . . . . . . . . . . . . 96 B Discussion of the closure modelling procedure presented by Lloyd et al. (2004) 97 C Evaluating the displacement ∆s. 99. D Derivation of the Ergun equation. 101. D.1 Blake-Kozeny equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 D.2 Carman-Kozeny-Blake equation . . . . . . . . . . . . . . . . . . . . . . . . 102 D.3 Burke-Plummer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 D.4 Ergun equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 E Generalized shear stress model. 105. E.1 Generalized plane Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . . . 106. vii.

(9) E.2 Reynolds number and friction factor for power law flow through granular porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 E.2.1 Reynolds number used by Smit (1997) . . . . . . . . . . . . . . . . 108 E.2.2 Reynolds number used by Smit & Du Plessis (2000) . . . . . . . . . 109 E.2.3 Friction factor used by Smit (1997) and Smit & Du Plessis (2000) . 109. viii.

(10) Nomenclature Standard characters a. [m−1 ]. specific surface. av. [m−1 ]. solid specific surface. A. []. coefficient in the Blake-Kozeny equation. Ap. [m2 ]. cross-sectional flow area. B. []. coefficient in the Burke-Plummer equation. d. [m]. linear dimension of RUC. dp. [m]. grain diameter. ds. [m]. linear dimension of solid cube in RUC. Dh. [m]. hydraulic diameter. Dp. [m]. spherical particle diameter. f. [m−2 ]. shear factor. fb. [N.kg −1 ]. external body forces per unit mass. F. []. dimensionless shear factor. F. [N ]. drag force −1. g. [N.kg ]. gravitational constant. k. [m2 ]. hydrodynamic or Darcy permeability. kkoz. []. Kozeny constant. ko. []. shape factor. K. [N.s.m−2 ]. consistency index of power law fluid. K. []. dimensionless hydrodynamic permeability. l. [m]. length scale of microscopic structure. L. [m]. length scale of macroscopic structure. n. []. behaviour index of power law fluid. n. []. inwardly directed unit vector normal to surface of solid ix.

(11) []. unit vector in streamwise direction. p. [P a]. interstitial pressure. q. [m.s−1 ]. superficial velocity, Darcy velocity or specific discharge. q mf. [m.s−1 ]. minimum fluidization velocity. Q. [m3 .s−1 ]. volumetric flow rate. ro. [m]. position vector of REV centroid. R. [m]. radius. Rh. [m]. hydraulic radius. Re. []. Rec. []. pore Reynolds number (ρ q (d − ds )/µ). Rep. []. Sff. [m2 ]. Sf s. [m ]. fluid-solid interface in RUC. Sfs. [m2 ]. fluid-solid interfaces in REV. Sg. [m2 ]. surface area in RUC adjacent to stagnant fluid volume. S||. [m2 ]. surface area in RUC adjacent to streamwise fluid volume. S⊥. [m2 ]. surface area in RUC adjacent to transverse fluid volume. Uf. [m3 ]. total fluid volume in RUC. Uf. [m3 ]. total fluid volume in REV. Ug. [m3 ]. total stagnant volume in RUC. Uo. [m3 ]. total (fluid and solid) volume of RUC. Uo. [m3 ]. total (fluid and solid) volume of REV. Us. [m3 ]. total solid volume in RUC. Us. [m3 ]. total solid volume in REV. Ut. [m3 ]. total transfer volume in RUC. U||. [m3 ]. total streamwise volume in RUC. U⊥. [m3 ]. total transverse volume in RUC. u. [m.s−1 ]. drift velocity. v. [m.s−1 ]. interstitial fluid velocity. wk. [m.s−1 ]. streamwise average pore velocity. w⊥. [m.s−1 ]. transverse average pore velocity. x, y, z. [m]. distance along Cartesian coordinate. b n. critical Reynolds number. particle Reynolds number (ρ q ds /µ) 2. fluid-fluid interfaces in REV. x.

(12) Greek symbols β. []. average pore velocity ratio. γ˙. [s−1 ]. shear rate. δ. []. change in transverse property. ∆. []. change in streamwise property. ǫ. []. porosity. ǫB. []. backbone porosity. ǫc. []. percolation threshold porosity. ǫmf. []. minimum fluidization porosity. η. [N.s.m−2 ]. apparent viscosity. Λ. [s−1 ]. dimensionless resistance factor. µ. [N.s.m−2 ]. fluid dynamic viscosity. ξ. []. shear stress reduction coefficient. ρ. [kg.m−3 ]. fluid density. τ. [N.m−2 ]. local shear stress. τw. [N.m−2 ]. local wall shear stress. φ. []. any tensorial fluid phase quantity. φs. []. sphericity factor. Φsg. []. total gas/particle drag coefficient. χ. []. tortuosity factor. ψ. []. geometric factor. Miscellaneous ∇. del operator. h i. phase average operator. {}. deviation operator. h if e. intrinsic phase average operator interchange in unit vectors vector (underlined) diadic (doubly underlined). xi.

(13) Acronyms REV. Representative Elementary Volume. RUC. Representative Unit Cell. Subscripts f. fluid matter. fs. fluid-solid interface. h. hydraulic. o. total solid- and fluid volume. s. solid matter. w. wall. ||. parallel to streamwise direction. ⊥. perpendicular to streamwise direction. 0. lower limit. 1. higher limit. xii.

(14) Chapter 1 Introduction The term granular porous medium refers to a material consisting of an unconsolidated solid matrix with interconnected pores, as illustrated in Figure 1.1. One or several solidand fluid phases may be involved. A porous medium is said to be permeable if it is possible for the fluid phase to traverse through the interconnected pore sections. The term permeability is therefore used to describe the extent of conductance of fluid flow through the porous medium. fluid phase. solid phase. Figure 1.1: A two-dimensional schematic representation of an unbounded granular porous medium. The solid arrows indicate the direction of fluid flow through the pores. A characteristic bulk property of a porous medium is the porosity ǫ which is defined as the ratio of the void (which may be filled with liquid or gas) volume to the total (void and solid) volume of the porous medium. Granular porous media are classified as unconsolidated media and occur either naturally or it is constructed commercially for various engineering applications. Examples are the natural phenomenon of fluid flow through granular soils such as sand, rock and sandstones and water seepage through the subsoil of dams and other construction materials. Sandstone is a natural porous rock formation of very low 1.

(15) porosity (0.02 < ǫ < 0.35). Granular packed beds (0.25 ≤ ǫ ≤ 0.47) and fluidized beds (0.35 ≤ ǫ ≤ 0.8) are utilized for various applications in the chemical, pharmaceutical and petroleum industries. Packed columns are widely used in fixed- and fluidized bed reactors, mass and heat transfer operations, separation processes and filtration. Many years of research have been devoted to predicting the permeability of low to moderate porosity granular porous media. The ability to accurately predict the permeability through any type of granular porous medium, depends on a detailed description of the granular microstructure. A thorough knowledge of the interstitial properties of the porous medium is, however, an arduous task to obtain due to the complex geometry of the porous matrix. As a result there are very few analytical models in the literature for predicting the permeability through granular porous media. The customary procedure to follow recently is to solve the interstitial momentum transport equations through numerical simulations. Instead one seeks simple analytical techniques for predicting the permeability as a function of the porosity without the need to obtain information on the complex interstitial properties of the porous medium.. 1.1. Granular models from literature. The methods for modelling flow through granular porous media found in the literature may be classified more or less into three categories, i.e. the capillary tube or hydraulic radius model, the submerged object model and models based on a statistical averaging approach. In the capillary tube or hydraulic radius approach the flow through a granular packed bed is regarded as being equivalent to flow through a network of capillary tubes of varying cross-section but with a constant average cross-sectional area. The velocity profile is obtained by solving the Navier-Stokes equation for steady and fully developed flow. An expression is obtained for the pressure drop prediction across the packed bed. Ergun (1952) proposed a semi-empirical capillary tube model for predicting the pressure drop of a Newtonian fluid across a packed bed for Reynolds numbers ranging from the laminar to the highly turbulent flow regimes. Despite many critical comments by many authors in the past (e.g. Dagan (1989) and Brea et al. (1976)) on the rather unrealistic capillary representation of a packed bed, the Ergun equation proves to be somewhat more successful than the submerged object models, based on the frequent use of the equation to serve as the onset of many other proposed models in the literature. The Ergun equation has been modified by many authors (e.g. Gidaspow (1994), Yu et al. (1968) and Mishra et al. (1975)) to predict the flow behaviour in a fluidized bed. Mehta & Hawley (1969) modified the Ergun equation to take into account the effect of the column wall when the column to particle diameter ratio is small. Bird et al. (2002) was the first to modify the Ergun equation to describe the rheological flow behaviour of non-Newtonian fluids in porous media. The Ergun equation has also been modified extensively (Christopher & Middleman (1965), Kemblowski & Michniewics (1979), Brea et al. (1976)) to account for 2.

(16) non-Newtonian power law flow. Macdonald et al. (1979) verified the predictive capabilities of the Ergun equation with experimental data from the literature involving granular porous media of various porous microstructures and proposed different coefficient values for the Ergun equation. Sabiri & Comiti (1995) and Chhabra & Srinivas (1991) investigated the flow of a non-Newtonian purely viscous power law fluid through granular beds experimentally and proposed an expression for predicting the pressure drop across the bed based on the capillary tube model. Chakrabarti et al. (1991) investigated the rheology of various concentrations of a commercial polymer solution through beds consisting of spherical particles experimentally by using the capillary tube model. The submerged object approach regards flow around an assemblage of submerged objects forming a spatial array. The customary procedure to follow in determining the drag force on a typical particle in the assemblage is by modification of the Stokes’ drag on a single particle to account for the additional resistance arising from the presence of neighbouring particles. Stokes’ flow (e.g. Chorlton (1967)) involves the steady creeping motion of an incompressible Newtonian fluid with a uniform approaching velocity past an isolated, stationary sphere embedded in a fluid of infinite extent. The drag force on the sphere is determined by solving the Stokes equations. Chester (1962) pointed out that when the Reynolds number is not negligibly small Stokes’ drag for flow past a sphere is inadequate since the inertial terms are not negligible at great distance from the sphere. The drag force obtained from solving Oseen’s equations provides a first order expansion of the Reynolds number. Hasimoto (1958) considered the steady motion of an incompressible Newtonian fluid past a periodic array of small particles in a dilute medium. The drag force on a typical sphere within the array was obtained by modification of the Stokes equations. Happel (1958) proposed a concentric spherical cell model for predicting the flow of a Newtonian fluid through a random assemblage of spheres of low porosity in the creeping flow regime. The assemblage of spheres is regarded as a periodic array of identical spherical cells. Each cell contains a single sphere surrounded by a fluid envelope with a frictionless boundary. The Stokes’ equations subjected to appropriate boundary conditions were solved and by application of Darcy’s law the pressure drop prediction across the bed was obtained. Various statistical averaging methods have been proposed in the literature, e.g. the method involving a spatial volume averaging over a representative elementary volume, the method of homogenization for application to Stokes’ flow through periodic structures and purely statistical averaging methods concerning probability density and uncertainty. 3.

(17) distribution functions. Dagan (1989) proposed a purely statistical model for predicting the permeability for steady flow of an incompressible Newtonian fluid through a granular porous medium of low porosity. The porous medium is regarded as a network of three-dimensional planar fissures with interconnected pores and identical constant apertures. Du Plessis (1994) proposed an analytical model for predicting the pressure gradient for Newtonian flow through granular porous media for all porosities and Reynolds numbers ranging from the Darcy regime to the steady laminar limit of the Forchheimer regime. The pore-scale model is based on a rectangular representation of the average granular porous microstructure. Smit & Du Plessis (2000) extended the rectangular representative unit cell model of Du Plessis & Masliyah (1991) to provide an analytical pressure drop prediction for non-Newtonian purely viscous power law flow through porous media of various types of porous microstructures, including granular media.. 1.2. Objective. The capillary tube models are semi-empirical models in which empirical factors are introduced for correlation with experimental data. Many of the submerged object models, on the contrary, are exact analytical models (e.g. the models of Stokes and Hasimoto) and therefore lack the ability to be generalized for a broader field of application. Although the capillary tube model have the ability to be extended for various other fields of applicability, its main draw back is its empiricism. Consequently, the need arises to produce a simple generic analytical model of which the assumptions made within the analytical modelling procedure may easily be adapted to broaden its range of applicability. Over the past two decades an analytical model has been developed at the University of Stellenbosch for predicting fluid flow through various types of porous media. The original model was proposed in 1988 and has ever since been adapted to improve its predictive capabilities. The objective of this work is to present the most recent improvement of the analytical model for predicting the pressure drop across a granular porous medium for Reynolds numbers within the steady laminar flow regime and over a wide range of porosities.. 4.

(18) 1.3. Assumptions. In order to provide a relatively simple, but still realistic, pore-scale model to approximate the complex geometry of the granular porous microstructure, some simplifying assumptions need to be made. This work concerns three-dimensional, isothermal, steady laminar flow of an incompressible viscous fluid through a granular porous medium. The porous medium is assumed to be homogeneous and isotropic with respect to the average geometrical properties. The porous medium is also assumed to be unbounded. Wall effects due to external boundaries may therefore be neglected and as a result the local porosity may be assumed to be constant. The pore sections are assumed to be inter-connected, but may contain stagnant regions where the fluid remains stationary. The solid constituents are assumed to be uniformly sized, rigid, smooth and randomly distributed in all directions. The traversing fluid is assumed to consist of a single fluid phase, i.e. only saturated fluid flow is considered, with constant physical properties, unless otherwise stated. Both phases will be treated as a continuum and therefore the terms ‘particle’ and ‘grain’ will be regarded as equivalent. It is furthermore assumed that the grains remain stationary, which may be justified by the fact that in a packed bed the grains are supported by inter-particle contact (Happel & Brenner (1965)).. 1.4. Layout of thesis. The commencement of the analytical model to be introduced is the method of volume averaging of the transport equations describing the motion of the fluid through the porous medium. The application of this method to the relevant transport equations is discussed in chapter 2. The granular pore-scale model is introduced in chapter 3 together with a discussion of the laminar flow regimes under consideration. Chapters 4 and 5 are devoted to the analytical modelling of the pore-scale model within the two asymptotic limiting flow regimes discussed in chapter 4. The pore-scale model for Newtonian flow is presented in chapter 6 and compared with the semi-empirical Ergun equation. The rest of this work concerns generalizations of the Newtonian model. In chapter 7 the model is adapted to account for non-Newtonian flow and in chapter 8 the Newtonian model is extended for application in fluidized beds and sandstones. Finally some conclusions are drawn in chapter 9.. 5.

(19) Chapter 2 Method of Volume Averaging This chapter concerns the local volume averaging of the momentum equations governing the fluid transport within the pores of the porous medium. The method of volume averaging provides a manner of relating the interstitial flow conditions to the measurable macroscopic flow behaviour.. 2.1. Interstitial transport equations. The governing equations describing single phase flow of an incompressible Newtonian fluid in an infinitely permeable porous medium are the continuity equation for conservation of mass, i.e. (2.1) ∇·v = 0, and the Navier-Stokes equation for interstitial momentum transport derived from Newton’s second law (e.g. Happel & Brenner (1965)) ρ. ∂v + ∇ · (ρ v v) = −∇P + ∇ · τ + ρ f b , ∂t. (2.2). where v is the interstitial fluid velocity at any point within the pore space, ρ is the constant fluid density, P is the interstitial hydrostatic pressure, τ is the local shear stress tensor of a Newtonian fluid and f b denotes the external body forces per unit mass. The terms on the left hand side of equation (2.2) represent the time rate of change in momentum per unit volume of fluid, constituting the inertial forces, and the terms on the right hand side represent the external forces per unit volume contributing to the nett force exerted on the differential fluid element. The external forces include body forces, e.g. gravitation, and pressure and viscous forces exerted on the surface of the fluid element. Assuming that the gravitational force is a conservative force and discarding other external forces, f b may be expressed as f b = −∇gz where z denotes an elevation and the gravitation 6.

(20) constant g is assumed to remain constant with variations in z. Under these conditions, the terms −∇P and ρf b may be combined to form a single term, i.e. −∇(P + ρ gz). For relatively small values of z the gravitational term may be assumed to be negligible. The pressure p = P + ρ gz will henceforth be referred to as the interstitial dynamic pressure. The Navier-Stokes equation may accordingly be expressed as ρ. ∂v + ∇ · (ρ v v) = −∇p + ∇ · τ , ∂t. (2.3). The method of volume averaging over a representative portion of the fluid domain to obtain local measurable macroscopic volume averaged transport equations has been studied by many authors (e.g. Bear & Bachmat (1986), Slattery (1969) and Whitaker (1969)). The next section will shortly address some of the basic principles on which the method is based.. 2.2. Representative Elementary Volume. A Representative Elementary Volume, abbreviated REV, is defined as an averaging volume U o of finite extent within the porous domain (e.g. Whitaker (1999)), consisting of both fluid and solid phases, respectively denoted by U f and U s . A two-dimensional schematic representation of a spherical REV is shown in Figure 2.1.. Sfs Us. L. ro. Uf. O. l. Sff. Figure 2.1: A two-dimensional schematic representation of a spherical REV. The dashed line indicates the REV boundary. An REV is defined at each an every point within the unbounded porous medium. The centroid of each REV is indicated by a position vector ro relative to some arbitrary origin 7.

(21) O, as illustrated in Figure 2.1 for a single REV. Inter-connectivity of the pore space and treating the fluid phase as a continuum are essential requirements for an REV. Although the shape of the REV is not prescribed, it should ensure that the averaging functions are continuous and also continuously differentiable to any order. It is however required that the size, shape and orientation of the REV should remain constant. The REV is chosen to be the smallest possible volume containing sufficient fluid and solid parts to be statistically representative of the local average properties, e.g. the local porosity. The size of the REV is appropriately chosen when small variations in the local volume will not change the values of the local average properties. In terms of the linear dimensions indicated in Figure 2.1, this will require that l >> dp and l << L. The fluid-solid interfaces within the REV are denoted by S f s and the fluid-fluid interfaces on the REV boundary by S f f . The porosity ǫ of the REV is assumed to be uniform and constant and is defined by the volumetric ratio Uf . (2.4) ǫ ≡ Uo. 2.3. Macroscopic volume averaged quantities. The concept of an REV leads to the introduction of various measurable macroscopic volume averaged quantities. The superficial velocity q, also known as the Darcy velocity or specific discharge, is defined as the phase average (Appendix A.1) of the interstitial fluid velocity v, i.e. 1 ZZZ vdU , q = hvi = (2.5) Uo Uf and represents the average velocity that would prevail in a section of the porous medium in which no solid phase is present. For this reason it is customary to use the superficial velocity in the comparison of different flow systems. The streamwise direction bn is defined as the direction of the superficial velocity, that is b n = q/q .. (2.6). The drift velocity u is defined as the intrinsic phase average (Appendix A.1) of the interstitial velocity v, i.e. 1 ZZZ u = h v if = vdU , (2.7) Uf Uf and represents the average fluid velocity in the streamwise direction. The relationship between the superficial- and drift velocity is given by q = ǫu ,. 8. (2.8).

(22) which is known as the Dupuit-Forchheimer relation. The deviation of any fluid phase tensorial quantity φ at any point within U f is denoted by {φ} and defined as {φ} ≡ φ − hφif .. 2.4. (2.9). Macroscopic transport equations. Volume averaging of the continuity equation for an incompressible fluid over a sufficiently large REV (Appendix A.2) leads to ∇·q = 0,. (2.10). and the volume averaged Navier-Stokes equation (Appendix A.3) for an incompressible fluid may be expressed as −∇ h p i = ρ. D E ∂q + ρ∇ · q q/ǫ + ρ∇ · h{v} {v}i − ∇ · τ ∂t 1 ZZ + n {p} − n · τ d S . Uo Sfs. (2.11). where n is the inwardly directed unit vector normal to the surface of the solid and h p i denotes the average macroscopic pressure. Equation (2.11) predicts the streamwise pressure gradient for unidirectional average flow over any type of porous medium, e.g. granular media, foams and fibre beds. The surface integral contains all the information on the fluidsolid interaction and depends strongly on the porous microstructure. The evaluation of the surface integral is subjected to a detailed and accurate description of the interstitial pressure- and velocity gradients at the fluid-solid interfaces. In order to circumvent the complex geometry of the solid constituents, a pore-scale model resembling the porous microstructure will be introduced in the following chapter to approximate and quantify the surface integral for the particular case of a granular porous medium.. 9.

(23) Chapter 3 Rectangular Granular Pore-Scale Model This chapter introduces the concept of a pore-scale model for closure modelling of the interstitial fluid-solid interaction to analytically quantify the pressure gradient prediction over a granular porous medium.. 3.1. Rectangular Representative Unit Cell. A rectangular Representative Unit Cell, abbreviated RUC, was originally introduced by Du Plessis & Masliyah (1988) for isotropic sponge-like media. A rectangular RUC is defined as the smallest rectangular control volume, Uo , into which the local average properties of the REV may be embedded. The granular RUC model was introduced by Du Plessis & Masliyah (1991) after which some of the model assumptions were improved by Du Plessis (1994). The latter model will henceforth be referred to as the existing RUC model. The granular RUC model is schematically illustrated in Figure 3.1. The fluid filled volume within the RUC is denoted by Uf and Us denotes the volume of the solid phase. The RUC is assumed to be homogeneous and isotropic in accordance with the average geometry of the porous medium. The assumption of average geometrical isotropy allows the introduction of a cubic RUC of linear dimension d, defined as the average length scale over which similar changes in geometrical and physical properties take place. The solid cube represents the average geometric properties of the granular solid microstructure. The length of the cube is denoted by ds . The cubic geometry of the RUC model is introduced for mathematical simplicity and serves as an approximation for flow through an assemblage of grains with arbitrary shape. The solid cube is assumed to be stationary and is positioned so that a vector normal to any of the cube’s faces is parallel to a normal vector on the corresponding face of the RUC. Due to the parallel alignment of the solid cubes’ faces with that of the RUC, one of the fluid channels will always be aligned with the streamwise direction, leaving the remaining two channels to be directed in transverse 10.

(24) d Uf d. ds. Us. b n. ds. ds d. Figure 3.1: Cubic geometry of the RUC model for modelling the fluid flow through an isotropic granular porous medium. The streamwise direction is indicated by bn.. directions, that means, directions perpendicular to the streamwise direction. Any property referring to the streamwise direction will henceforth be denoted by a subscript k and any property related to a transverse direction by ⊥. The porosity ǫ of the RUC is assumed to be equivalent to the porosity of the REV and is defined as ǫ =. Uf . Uo. (3.1). The fluid filled volume Uf and the volume of the solid cube Us may be expressed in terms of the linear dimensions d and ds as follows Uf = ǫ Uo = ǫ d3 ,. (3.2). Us = d3s ,. (3.3). which lead to the following relationship between the linear dimensions of the RUC in terms of the porosity ǫ: 1 (3.4) ds = (1 − ǫ) 3 d .. 3.2. Staggered and non-staggered arrays. The relative transverse positioning of neighbouring RUC’s in the streamwise direction leads to the introduction of two arrays yielding different fluid flow phenomena. The array in which maximum possible staggering of the solid cubes in a straight streamtube of width d occurs in the streamwise direction, is referred to as a fully staggered array and the array in which no staggering occurs in any of the three principal directions, is referred to as a 11.

(25) regular array. This section forms an extension of the existing RUC model since the latter model did not consider an array in which splitting of the stream-tube occurs neither one in which stagnant regions are present.. 3.2.1. Fully staggered array. A typical two-dimensional RUC within a streamwisely fully staggered array is schematically illustrated in Figure 3.2.. b n. Figure 3.2: A two-dimensional schematic representation of a fully staggered array. The boundaries of a typical RUC is indicated by the bold dashed lines and the dotted lines represent a stream-tube. In a fully staggered array the streamwise flux is split and then deviated in opposite transverse directions to traverse past the solid cube opposing the motion of the fluid in the streamwise direction. The streamwise flux reunites on the lee side of this cube and proceeds to flow in the streamwise direction before the next solid obstacle causes the streamwise flux to split again and the process repeats itself. A fully staggered array contains no stagnant regions and no staggering occurs in the two transverse directions. Also shown in Figure 3.2 is a stream-tube, represented by the dotted lines, that serves as a fluid envelope through which the fluid flows in the streamwise direction. Since the stream-tube consists of a bundle of streamlines which may not cross, it is assumed that all the fluid enters the RUC through the upstream face and exits through the downstream face with no fluid exchange across the transverse facing surfaces of the RUC. Figure 3.3 is a three-dimensional schematic representation of an RUC within a fully staggered array.. 12.

(26) ds 2. b n. Us. ds. d Figure 3.3: A three-dimensional schematic representation of an RUC within a fully staggered array. Figure 3.4 represents a two-dimensional upstream view of the RUC associated with a fully staggered array. It may be visualized that the fluid is flowing out of the paper around the cube at the rear and then transversally to exit past the forward quarter cubes.. d − ds. d. ds 2. d Figure 3.4: A two-dimensional upstream view of an RUC associated with a fully staggered array.. 3.2.2. Regular array. Figure 3.5 is a two-dimensional schematic representation of a typical RUC within a regular array. Also shown is the stream-tube associated with such an array.. 13.

(27) b n. Figure 3.5: A two-dimensional schematic representation of a regular array. The boundaries of a typical RUC is indicated by the bold dashed lines and the dotted lines represent a stream-tube. In a regular array the fluid enters and leaves the RUC in the streamwise direction without being deviated in a transverse direction, that is, in a regular array no staggering occurs. A regular array contains stagnant regions where the fluid remains stationary between any two neighbouring cubes in the streamwise direction. Note that in each of the three principal directions normal to the cube faces no staggering occurs. A three-dimensional schematic representation of an RUC within a regular array is shown in Figure 3.6.. ds. b n. Us. d Figure 3.6: A three-dimensional schematic representation of an RUC within a regular array. Figure 3.7 shows a two-dimensional upstream view of a regular array. The fluid may be visualized to be flowing out of the paper, past the cubes which are positioned directly behind each other in the streamwise direction.. 14.

(28) d − ds d ds. d Figure 3.7: A two-dimensional upstream view of an RUC associated with a regular array.. 3.3. Piece-wise straight streamlines. The adopted rectangular geometry and the isotropy requirement of the RUC model allow for the facing surfaces of any two neighbouring cubes to be a uniform distance d − ds apart, yielding pair-wise sets of equal parallel plates. In conjunction with the parallel plate configuration, piece-wise straight streamlines, between and parallel to the plates, are assumed to prevail within all the flow channels, as illustrated in Figure 3.8 for the respective arrays.. b n. (a) Fully staggered array. (b) Regular array. Figure 3.8: A two-dimensional representation of the piece-wise straight streamlines associated with (a) a fully staggered array and (b) a regular array. The dashed lines represent the streamlines and the bold dashed lines a typical RUC. The splitting of the stream-tube into two equal but directionally opposite transverse fluid volumes within a fully staggered array is clearly illustrated by the streamlines in Figure 3.8 (a). Figure 3.8 (b) shows the stagnant volume, indicated by the absence of streamlines, present between any two neighbouring cubes in the streamwise direction within a regular array. 15.

(29) 3.4. Volume partitioning. The piece-wise straight streamlines allow for the fluid domain within an RUC to be partitioned into different sub-volumes depending on the orientation of the particular fluid volume with respect to the streamwise direction and the presence of surfaces adjacent to the specific fluid volume under consideration. The concept of volume partitioning of the fluid domain was not considered in the existing RUC model. The volume partitioning of the fluid domain within the RUC presented in Figure 3.6 for a regular array is shown in Figure 3.9. As opposed to the other figures shown thus far, the shaded volumes within Figure 3.9 represent fluid volumes. The solid volumes are not shown.. Sk. Uk. Ut Uk. b n. Ug. Sg. b n. Figure 3.9: A three-dimensional volume partitioning of the fluid domain within the RUC presented in Figure 3.6 for a regular array. The shaded volumes represent fluid volumes, solid volumes are not shown. The fluid volume in the channels parallel to the streamwise direction and adjacent to solid surfaces is denoted by Uk and is referred to as a streamwise volume. The solid surfaces adjacent to Uk are denoted by Sk and are referred to as the streamwise surfaces. The fluid volume in the stagnant regions between any two neighbouring cubes in the streamwise direction, is denoted by Ug and is referred to as a stagnant volume. The solid surfaces adjacent to Ug are denoted by Sg and are referred to as the stagnant surfaces. The fluid volume involving no shear stresses due to the absence of adjacent solid surfaces is denoted by Ut and is referred to as a transfer volume. Similarly, volume partitioning may be applied to the fluid domain within the RUC presented in Figure 3.3 for a fully staggered array. The volume partitioning of a fully staggered array is not shown because of the complexity of illustrating it graphically. As opposed to the stagnant region within a regular array, a fully staggered array is characterized by a fluid volume in which the fluid flows in directions perpendicular to the streamwise direction. These fluid volumes, denoted by U⊥ , are adjacent to solid surfaces and are referred to as the transverse volumes. The solid surfaces adjacent to U⊥ are denoted by S⊥ and are referred to as the transverse surfaces. The total fluid volume Uf contained within an RUC associated with any of the two arrays may thus be expressed as Uf = Uk + U⊥ + Ug + Ut , 16. (3.5).

(30) and the total fluid-solid interfaces Sf s may accordingly be expressed as Sf s = Sk + S⊥ + Sg .. (3.6). For a fully staggered array Ug = Sg = 0 and for a regular array U⊥ = S⊥ = 0. The expressions for the three-dimensional surface- and volume partitioning of the respective arrays are presented in Table 3.1 in terms of the linear dimensions d and ds . The expressions presented in Table 3.1 denote the total volume (or surface area) associated with the specified volume (or surface).. Array Parameter Fully staggered Uo. d3. Us. d3s. Uf. d3 − d3s. Ut. (d − ds )2 (d + 2ds ) 2 d2s (d − ds ). Uk. 4 d2s. Sk U⊥. Regular. d2s (d − ds ). 0. S⊥. 2 d2s. 0. Ug. 0. d2s (d − ds ). Sg. 0. 2 d2s. Table 3.1: Three-dimensional surface- and volume partitioning for the RUC’s associated with a fully staggered- and regular array.. The volume partitioning of the fluid domain allows for the introduction of a streamwise average pore velocity wk (Du Plessis (1994)), defined as ZZZ 1 wk = v dU Uk + Ut Uf. (3.7). and relates as follows to the superficial velocity q, wk =. q d2 , Ap k. 17. (3.8).

(31) where Apk is the streamwise cross-sectional flow area available for fluid discharge through the RUC, i.e. Apk = d2 − d2s . (3.9) It thus follows that wk =. q d2 . d2 − d2s. (3.10). As listed in Table 3.1, different expressions are obtained for Uk and U⊥ implying that the average pore velocities within these two fluid volumes should differ. A coefficient β is therefore introduced to account for the different average velocities in the streamwise and transverse channels and is defined as w⊥ , (3.11) β≡ wk with w⊥ the magnitude of the transverse average pore velocity. A geometric factor ψ, defined as Uk + Ut + U⊥ + Ug Uf ψ = = , (3.12) Uk + Ut Uk + Ut was introduced by Lloyd et al. (2004). This factor yields the same result for a both a fully staggered- and non-staggered array, i.e. ǫ ψ = . (3.13) (1 − (1 − ǫ)2/3 ) The reason for the introduction of the geometric factor ψ will be addressed in the following chapter.. 3.5. Volume averaging of transport equations over an RUC. Volume averaging of the Navier-Stokes equation for incompressible flow over a typical RUC may be expressed as −∇ hpi = ρ. D E ∂q + ρ ∇ · q q/ǫ + ρ∇ · h{v} {v}i − ∇ · τ ∂t 1 ZZ + n {p} − n · τ dS , Uo Sf s. (3.14). where n denotes the inwardly directed unit vector normal to one of the cube faces. Together with the volume averaged continuity equation of an incompressible fluid, that is, ∇·q = 0, 18. (3.15).

(32) these equations describe the fluid transport through an RUC, which represents the average geometric and physical properties of the granular porous medium. Equation (3.14) provides an expression for the streamwise pressure gradient over the linear dimension d of the RUC. The momentum dispersion term, ρ∇ · h{v}{v}i, may be expressed as follows in terms of the superficial velocity q (Appendix A.3): D. E. ρ∇ · h{v}{v}i = ρ∇ · q q −. D E D E 2 1 ρ∇ · q q + 2 ρ∇ · q q , ǫ ǫ. (3.16). It thus follows that −∇ hpi = ρ. D E D E D E ∂q 2 1 + ρ ∇ · q q/ǫ + ρ∇ · q q − ρ∇ · q q + 2 ρ∇ · q q ∂t ǫ ZZ ǫ D E 1 n {p} − n · τ dS . −∇· τ + Uo Sf s. (3.17). D E. For a Newtonian fluid of constant viscosity µ, it follows that ∇ · τ = µ∇2 q, yielding −∇ hpi = ρ. D E D E D E ∂q 2 1 + ρ ∇ · q q/ǫ + ρ∇ · q q − ρ∇ · q q + 2 ρ∇ · q q ∂t ǫ ZZ ǫ 1 − µ∇2 q + n {p} − n · τ dS . Uo Sf s. (3.18). All the terms in equation (3.18), except the surface integral, are macroscopic terms. The term −µ∇2 q represents the macroscopic viscous shear at external walls, which may be neglected since the porous medium is assumed to be unbounded. Justified by approximated experimental conditions (Dybbs & Edwards (1982)), a uniform superficial velocity field q may be assumed. Since a uniform average velocity field implies macroscopic conservation of momentum, all the terms resulting from the interstitial rate of change in momentum should vanish macroscopically, which is indeed the case when q is assumed to be uniform in equation (3.18). It thus follows that for steady flow of an incompressible viscous fluid through a homogeneous porous medium in which a uniform average velocity field is assumed, the streamwise pressure gradient may be expressed as −∇ hpi =. 1 ZZ n p − n · τ dS . Uo Sf s. (3.19). Equation (3.19) represents a force balance between the external pressure gradient for fluid transport through the RUC and the pressure and viscous forces exerted by the solid cube on the traversing fluid. The first term in the surface integral denotes the inertial pressure forces exerted by the solid cube on the traversing fluid due to a pressure variation over the upstream and downstream facing surfaces of the cube. The pressure variation results 19.

(33) from a change in momentum across the streamwise facing surfaces of the cube. Note that these inertial pressure forces are interstitial forces which contribute to the external pressure gradient, as opposed to the macroscopic inertial forces that vanished due to the assumption of a uniform average velocity field. Interstitially changes in momentum occur, which become significant at higher Reynolds numbers, but macroscopically momentum is conserved. The second term in the surface integral denotes the viscous forces exerted by the solid cube on the traversing fluid due to shear stresses at the fluid-solid interfaces. Since the Reynolds number is defined as the ratio of the inertial forces to the viscous forces, the contribution of the pressure- and viscous forces to the streamwise pressure gradient depends on the magnitude of the Reynolds number. The Reynolds number used in the RUC model is a particle Reynolds number Rep , defined in terms of the length ds of the solid cube, i.e. ρ q ds . (3.20) Rep = µ The following chapters concern closure modelling of the surface integral of equation (3.19) in order to analytically quantify the fluid-solid interaction in the limit of very low Reynolds number flow and in the steady laminar limit of the inertial flow regime to obtain a general expression for the streamwise pressure gradient over a wide range of Reynolds numbers through application of an asymptote matching technique.. 3.6. Classification of laminar flow regimes. Three laminar flow regimes will be considered, namely the asymptotic limit of low Reynolds number flow in the Darcy or creeping flow regime, the steady laminar limit of the inertial or Forchheimer flow regime and the transition regime in between which is characterized by the development of boundary layers. The steady laminar limit is followed by an unsteady laminar flow regime and at even higher Reynolds numbers the boundary layer becomes turbulent (Dybbs & Edwards (1982)). Since only steady flow is considered, the latter two regimes fall beyond the scope of this work.. 3.6.1. Limit of low Reynolds number flow. The Darcy or creeping flow regime corresponds to a pore Reynolds number Re < 1 (Dybbs & Edwards (1982)). The analytical modelling procedure of the RUC model in this regime will concern pore Reynolds numbers within the asymptotic limit, i.e. Re ≈ 0.1. The analogue between the streamlines associated with flow past a sphere and a cube in the limit of low Reynolds number flow is schematically shown in Figure 3.10. The piece-wise straight streamlines assumed by the RUC model for flow past a cube is clearly illustrated.. 20.

(34) Flow past a sphere (Re ≈ 0.1). Flow past a cube (Re ≈ 0.1). Figure 3.10: A two-dimensional schematic representation of the streamlines associated with flow past a sphere and a cube in the limit of low Reynolds number flow. In this limit the viscous forces predominate over the inertial pressure forces and consequently the flow pattern is strongly influenced by the granular microstructure. The advantage of modelling fluid flow through the pores within this limit is that a fully developed velocity profile may be assumed to prevail throughout all the pore sections. The entrance effects arising from the gradual build-up of a developing velocity profile may thus be neglected.. 3.6.2. Transition regime. The transition from the Darcy to the Forchheimer regime is due to the development of boundary layers near the solid surfaces within the pores. This regime is associated with 1 < Re < 100 (Dybbs & Edwards (1982)). In the transition regime both the inertial and viscous forces contribute to the streamwise pressure gradient. The flow conditions in this regime are not modelled explicitly as it would require an enormous computational effort. The applied asymptote matching is assumed to be reasonably accurate in predicting the transitional effects.. 3.6.3. Steady laminar limit of the inertial flow regime. The steady laminar limit of the inertial flow regime corresponds to Re > 100 (Dybbs & Edwards (1982)). At Reynolds numbers just before the commencement of the inertial flow regime, the boundary layer begins to separate from the downstream stagnation point on the lee side of the solids grains. The boundary layer moves further downstream as the Reynolds number increases until the steady laminar limit is reached at which a separation zone is formed on lee side of the solids grains. The analogue between the separation zone for flow past a sphere and a cube within the steady laminar limit of the inertial flow regime is schematically illustrated in Figure 3.11.. 21.

(35) Flow past a sphere (Re > 100). Flow past a cube (Re > 100). Figure 3.11: A two-dimensional schematic representation of the streamlines associated with flow past a sphere and a cube in the steady laminar limit of the inertial flow regime. In the separation zone a low fluid velocity persists whereas, on the boundary of the separation zone, the fluid velocity is relatively high. As a result of the significant difference in flow velocities an interstitial recirculation pattern is generated within the separation zone. The entire square surface on the lee side of the solid is therefore exposed to a relatively low pressure. The resulting streamwise pressure difference between the upstream and downstream facing surfaces of the solid grains depends on the size of the separation zone which, in turn, depends on the position of the point of separation. The point of separation is again determined by the shape or form of the solid obstacle. This is the reason why the pressure force resulting from the pressure difference across a single isolated solid grain is referred to in the literature as form drag. For a laminar boundary layer separation occurs about midway between the front and rear of the solid (Roberson & Crowe (1985)). As a result of the separation of the boundary layer from the surface of the solid, the surface area on which the shear stresses act, is diminished substantially. Consequently, in the steady laminar limit of the inertial flow regime the pressure forces predominate over the viscous shear forces.. 22.

(36) Chapter 4 Low Reynolds Number Flow Regime This chapter involves the analytical closure modelling of the fluid-solid interaction within the RUC for predicting the streamwise pressure gradient in the asymptotic limit of low Reynolds number flow.. 4.1. Closure modelling at low to moderate porosities. The assumption of flow between parallel plates is only valid for low to moderate porosities where neighbouring cubes are present. At high porosities neighbouring cubes are absent so that the parallel plate configuration no longer persists and the assumption of flow between parallel plates is no longer valid. The RUC model within the limit of low Reynolds number flow will therefore be classified as a low to moderate porosity model (i.e. ǫ < 0.8). Fully developed laminar flow of a Newtonian fluid is assumed to prevail piece-wise throughout all pore sections. The closure modelling procedure with the three-dimensional granular RUC model to be presented in this work closely follows the work of Lloyd et al. (2004) for two-dimensional Newtonian flow perpendicular to a unidirectional fibre bed, although some discrepancies occur (Appendix B). The streamwise pressure gradient resulting from volume averaging of the transport equations in which a uniform velocity field q is assumed (Chapter 3), is given by −∇ hpi =. 1 ZZ n p − n · τ dS . Uo Sf s. (4.1). It was established by Lloyd (2003) that for low porosity media in the creeping flow regime the pressure term in the surface integral of equation (4.1) contains a viscous effect which contributes significantly to the streamwise pressure gradient. Although the viscous forces predominate over the inertial pressure forces in this regime, the pressure term should not be neglected. The shear stresses on the surfaces within the transverse channels are accompanied by transverse pressure gradients. These pressure gradients are contained within 23.

(37) the pressure term in the surface integral of equation (4.1). The commencement of the closure modelling procedure is the evaluation of the surface integral over the streamwise-, transverse- and stagnant surfaces within and adjacent to the RUC, yielding −∇ hpi =. ZZ 1 1 ZZ 1 ZZ n p dS + n p dS − n · τ dS Uo Uo Uo Sk S⊥ + Sg Sk ZZ ZZ 1 1 − n · τ dS − n · τ dS . Uo Uo S⊥ Sg. (4.2). The parallel alignment of the streamwise surfaces of neighbouring cubes in the transverse directions results in a vectorial cancelation of the pressure on the streamwise surfaces, that is, 1 ZZ (4.3) n p dS = 0 . Uo Sk Although streamlines may appear in the stagnant volume Ug of a regular array, the corresponding velocities will be very small (Lloyd (2003)). Assuming therefore that the shear stresses on Sg are negligible, it follows that 1 ZZ n · τ dS = 0 . Uo Sg. (4.4). Equation (4.2) there-upon simplifies to −∇ hpi =. 1 Uo. ZZ. S⊥ + Sg. n p dS −. 1 ZZ 1 ZZ n · τ dS − n · τ dS . Uo Uo S⊥ Sk. (4.5). The quasi-periodic structure of the RUC model in the streamwise direction, i.e. each upward transverse channel is to be followed by an opposite downward transverse channel, plays an important role in the further analysis of the remaining surface integrals regarding the particular location of the RUC in the streamwise direction. It is therefore of utmost importance that all possible locations of the RUC in the streamwise direction should be considered. The two typical choices of RUC’s chosen to accomplish the latter requirement are illustrated in Figures 4.1 and 4.2 for a fully staggered- and regular array, respectively. The upstream and downstream facing surfaces of the RUC with corner points AAAA cut through solid parts and the upstream and downstream facing surfaces of the RUC with corner points BBBB do not cut through any solid parts. S⊥AA and S⊥BB respectively denotes the fluid-solid interfaces adjacent to U⊥ of the RUC with corner points AAAA and BBBB in the fully staggered array shown in Figure 4.1. Similarly, SgAA and SgBB respectively denotes the fluid-solid interfaces adjacent to Ug of the RUC with corner points AAAA and BBBB in the regular array shown in Figure 4.2. 24.

(38) A. B. A. B. U⊥. U⊥ b n. U⊥ A. B. U⊥. A. B. Figure 4.1: Schematic illustration of the two typical choices of RUC’s in a fully staggered array to consider all possible locations in the streamwise direction. A. B. A. B. Ug. B. A. Ug. b n. B. A. Figure 4.2: Schematic illustration of the two typical choices of RUC’s in a regular array to consider all possible locations in the streamwise direction. The relative frequency of occurrence of the RUC with corner points AAAA in the streamwise direction is ds /d and the relative frequency of occurrence of the RUC with corner points BBBB in the streamwise direction is (d − ds )/d. If the transverse and stagnant surface integrals of equation (4.5) are weighed according to these relative frequencies of occurrence, over a streamwise displacement d, it follows that −∇ hpi =. ds 1 · d Uo. ZZ. n p dS +. d − ds 1 · d Uo. S⊥AA + SgAA ds 1 − · d Uo. ZZ. ZZ. n p dS. S⊥BB + SgBB. d − ds 1 n · τ dS − · d Uo. S⊥AA. ZZ. S⊥BB. 1 ZZ n · τ dS − n · τ dS . Uo Sk. (4.6). Assuming that the pressure p on all facing pairs of transverse and stagnant surfaces are 25.

(39) equal, results in a vectorial cancelation of the pressures on these surfaces, that is ZZ. ds 1 · d Uo. n p dS = 0 .. (4.7). S⊥AA + SgAA The RUC with corner points BBBB in the fully staggered array contains two transverse channels in which the fluid flows in opposite directions perpendicular to the streamwise direction. The shear stresses opposing the motion of the fluid on the surfaces adjacent to fluid volumes in these channels cancel vectorially. It thus follows that d − ds 1 ZZ · n · τ dS = 0 . d Uo S⊥BB. (4.8). Equation (4.6) there-upon simplifies to d − ds 1 −∇ hpi = · d Uo. ZZ. S⊥BB + SgBB. ds 1 ZZ n p dS − · n · τ dS d Uo S⊥AA 1 ZZ − n · τ dS . Uo Sk. (4.9). Defining τw as the magnitude of the local shear stress tensor τ on the solid surfaces, i.e. the wall shear stress, leads to d − ds 1 · −∇ hpi = d Uo. ZZ. S⊥BB + SgBB. ds 1 ZZ n p dS + · τw dS d Uo S⊥AA 1 ZZ + τw dS . Uo Sk. (4.10). Since fully developed flow is assumed within the asymptotic limit of low Reynolds numbers, the wall shear stress τw is assumed to be uniform and piece-wise constant over all the fluid-solid interfaces Sf s . The assumption of flow between parallel plates an equal distance d − ds apart allows for the interstitial velocity field within each channel section to be described by plane Poiseuille flow. The fully developed parabolic velocity profile associated with plane Poiseuille flow, together with the piece-wise straight streamlines assumed by the RUC model are illustrated in Figure 4.3 for a fully staggered- and regular array.. 26.

(40) b n. (a) Fully staggered array. (b) Regular array. Figure 4.3: A two-dimensional schematic representation of the fully developed parabolic velocity profile together with the piece-wise straight streamlines associated with (a) a fully staggered and (b) a regular array. Only half of the neighbouring solid volumes are shown for clearer representation of the fluid channels. Let τwk denote the wall shear stresses on the streamwise surfaces Sk and τw⊥ denote the wall shear stresses on the transverse surfaces S⊥ . The wall shear stress of a Newtonian fluid on the streamwise surfaces resulting from plane Poiseuille flow may be expressed as τ wk =. 6 µ wk . d − ds. (4.11). In a fully staggered array the streamwise flux divides to circumvent the solid obstacles causing the staggering in the streamwise direction. As a result the fluid flows in opposite directions within the transverse channels leading to a reduction in the wall shear stresses in the transverse channels of a fully staggered array due to the splitting of the streamtube. The latter effect is accounted for by the introduction of a coefficient ξ. The wall shear stresses on the transverse surfaces may accordingly be expressed as τw⊥ = β ξτwk =. 6 µ β ξ wk , d − ds. (4.12). where β is defined as in equation (3.11). The streamwise pressure gradient there-upon yields −∇ hpi =. d − ds 1 · d Uo. ZZ. n p dS +. S⊥BB + SgBB. τwk Sk + (ds /d) τw⊥ S⊥ b n. Uo. (4.13). In the limit of low porosities, that is, where the assumptions of the RUC model are satisfied, ds ≈ 1. (4.14) d 27.

(41) It thus follows that d − ds 1 · −∇ hpi = d Uo. ZZ. n p dS +. S⊥BB + SgBB. Sk + β ξ S⊥ τwk bn . Uo. (4.15). The remaining surface integral was shown by Lloyd (2003), to be expressible in terms of the gradient of the average pressure. The same procedure as for two-dimensional arrays of squares can be applied to three-dimensional arrays in the following way: Splitting the surface integral of equation (4.15) into one applicable to a fully staggered array and the other applicable to a regular array, yields d − ds 1 · d Uo. ZZ. n p dS =. S⊥BB + SgBB. d − ds 1 ZZ · n p dS + d Uo S⊥BB d − ds 1 ZZ · n p dS . d Uo SgBB. (4.16). The surface integral applicable to a fully staggered array may be expressed as " # (d − ds ) d2s d − ds 1 ZZ · n p dS = (∆p + δp) bn d Uo d Uo S⊥BB. U⊥ + Ug ∇ hpi Uf ! Uk + Ut − 1 ∇ hpi , Uf. = − =. (4.17). where ∆p is the total change in pressure in the streamwise volume and δp is the total change in pressure in the transverse volume. The above result (Lloyd (2003)) is obtained from the fact that ∆p + δp Uo b n = − ∇ hpi . (4.18) d Uf The surface integral applicable to a regular array may be expressed as " # d − ds 1 ZZ (d − ds ) d2s · ∆p bn n p dS = d Uo d Uo SgBB. U⊥ + Ug ∇ hpi Uf ! Uk + Ut − 1 ∇ hpi , Uf. = − = 28. (4.19).

(42) resulting from the fact that. ∆p Uo b n = − ∇ hpi . d Uf. (4.20). The streamwise pressure gradient applicable to both a fully staggered and a regular array may thus be expressed as −∇ hpi =. !. Uk + Ut Sk + β ξ S⊥ − 1 ∇ hpi + τwk bn , Uf Uo. (4.21). or after simplification −∇ hpi =. Uf Uk + Ut. !. Sk + β ξ S⊥ τwk bn . Uo. (4.22). The streamwise pressure gradient may there-upon be expressed as Sk + β ξ S⊥ U0 Sk + β ξ S⊥ = U0 Sk + β ξ S⊥ = d3 Sk + β ξ S⊥ = d. −∇ hpi =. ψ τwk bn. (Eqn. (3.12)). . . 6 µ wk b n (Eqn. (4.11)) d − ds 6 d2 µ q ψ (Eqn.(3.10) , U0 = d3 ) (d − ds )(d2 − d2s ) 6µq ǫ (Eqn. (3.13)) (1 − (1 − ǫ)2/3 ) (d − ds )(d2 − d2s ) d2s µ q ǫ = 12 (2 + β ξ) (Sk = 4d2s , S⊥ = 2d2s ) 2/3 2 2 (1 − (1 − ǫ) ) d (d − ds ) (d − ds ) ǫ (1 − ǫ)4/3 12 (2 + β ξ) (Eqn. (3.4)) (4.23) = 2 µq . d2s (1 − (1 − ǫ)1/3 ) (1 − (1 − ǫ)2/3 ) ψ. For a porous medium of local uniform porosity ∇ hpi = ǫ∇ hpif .. (4.24). A well-known expression for relating the external pressure drop to the superficial velocity is Darcy’s empirical law for steady unidirectional creeping flow of a Newtonian fluid through an unbounded isotropic granular porous medium of uniformly sized particles, i.e. ∆p µ = q, L k. (4.25). where ∆p is the pressure drop across the porous medium of length L and k is the Darcy or hydrodynamic permeability. For unidirectional flow in the positive x-direction of a Cartesian coordinate system, the streamwise pressure drop for Newtonian flow may, analogously to Darcy’s equation, be expressed as −. dp = µf00 q , dx 29. (4.26).

(43) where f00 is the shear factor of which the first subscript denotes the asymptotic limit of low Reynolds number flow and the second subscript denotes the asymptotic limit of low porosity. It thus follows that the shear factor f00 relates to the hydrodynamic permeability, which is dependent on the porous microstructure, as follows f00 =. 1 . k. (4.27). The shear factor f00 may consequently be expressed as f00 =. 12 (2 + β ξ)(1 − ǫ)4/3 2 . d2s (1 − (1 − ǫ)1/3 ) (1 − (1 − ǫ)2/3 ). (4.28). The dimensionless shear factor F00 = f00 d2s , describing the flow of a Newtonian fluid through both a fully staggered- and a non-staggered regular array, is thus given by 12 (2 + β ξ)(1 − ǫ)4/3 2 , (1 − (1 − ǫ)1/3 ) (1 − (1 − ǫ)2/3 ). F00 =. (4.29). or in terms of the dimensionless hydrodynamic permeability K = k/d2s , K =. . 1 − (1 − ǫ)1/3. . 1 − (1 − ǫ)2/3. 12 (2 + β ξ)(1 − ǫ)4/3. 2. .. (4.30). The transverse pore sections of a fully staggered array, in which the fluid flows in opposite directions, are assumed to be equal in volume. Therefore it is assumed that, similarly as in the case of two-dimensional flow, ξ = 21 for a fully staggered array. For a regular array ξ = 1, since in a regular array no splitting of the streamtube occurs. Due to the fact that a regular array does not contain any transverse channels, it follows that for a regular array β = 0. The determination of the value of β for a fully staggered array requires further analysis and will be addressed in the following subsection.. 4.1.1. Evaluation of the coefficient β. The RUC associated with a fully staggered array can be partitioned into four symmetric parts, as illustrated in Figure 4.4. Consider one of the four symmetric parts of the RUC presented in Figure 4.4. The bottom left hand part is arbitrarily chosen for illustration purposes. Let ∆s denote the displacement of the centroid of the fluid due to the shifting thereof to circumvent obstacles for streamwise discharge. Figure 4.5 illustrates the transverse displacement ∆s for fluid discharge through the RUC corresponding to the bottom left hand part of Figure 4.4. Let ∆x and ∆y respectively denote the transverse displacement of the centroid of the fluid in the x- and y-directions of a Cartesian coordinate system, as shown in Figure 4.5. 30.

(44) Figure 4.4: A two-dimensional illustration of the partitioning of the RUC associated with a fully staggered array into four symmetric parts.. ds 2. ∆s. ∆y. d 2. ∆x y x. Figure 4.5: Illustration of the transverse displacement ∆s of the centroid of the fluid for discharge through the RUC corresponding to the bottom left hand part of the RUC shown in Figure 4.4. The fluid volume in the transverse channels U⊥ may be expressed as U⊥ = Ap⊥ ∆s ,. (4.31). where Ap⊥ denote the cross-sectional flow area available for fluid discharge through the transverse channels within the RUC. Conservation of mass requires that Q = w k Ap k = w ⊥ Ap ⊥ ,. (4.32). where Q denotes the volumetric flow rate. From equations (4.31), (4.32) and (3.11) it follows that β may be expressed as β =. Apk ∆s . U⊥ 31. (4.33).

(45) The displacement ∆s may be obtained by determining the position of the centroid of the fluid before and after the transverse shift (Appendix C), yielding √ 2 2 ds ∆s = . (4.34) d + ds It thus follows that the value of β for a fully staggered array, yields √ √ (d2 − d2s ) 2 d2s 2. β = = d2s df d + ds. (4.35). Since each of the four symmetric parts of the RUC shown in Figure 4.4, yields √ the same value for β and contributes evenly to the nett effect of the flow field, β = 2 may be taken as the average value for β for a fully staggered array.. 4.1.2. Isotropic RUC model. The resulting expressions for the dimensionless shear factor F00 and the dimensionless hydrodynamic permeability K for a fully staggered- and a regular array are presented in Table 4.1.. Array. F00 4/3. Fully staggered Regular. 32.5 (1 − ǫ) 2 (1 − (1 − ǫ)1/3 ) (1 − (1 − ǫ)2/3 ) 24 (1 − ǫ)4/3 2 (1 − (1 − ǫ)1/3 ) (1 − (1 − ǫ)2/3 ). . K. 1 − (1 − ǫ)1/3. . 2. 1 − (1 − ǫ)1/3. . 2. 1 − (1 − ǫ)2/3. 32.5 (1 − ǫ)4/3. 1 − (1 − ǫ)2/3. 24 (1 − ǫ)4/3. Table 4.1: Expressions for the dimensionless shear factor F00 and the dimensionless hydrodynamic permeability K for a staggered and non-staggered array. It should be noted that the RUC model is a theoretical model which provides a simple hypothetical approximation of the actual complex granular porous microstructure. The RUC can not be regarded as a repetitive building block, since it is practically impossible to construct an isotropic array. An isotropic medium requires the random distribution of a large number of grains to obtain staggering in all directions. The construction of a streamwisely staggered array results in non-staggeredness in the other two principle directions perpendicular to the streamwise direction. Neither a fully staggered array nor a regular array is therefore isotropic. A fully staggered array is staggered only in the streamwise direction. In the other two principal directions perpendicular to the streamwise direction no staggering occurs, which corresponds to a regular array. The RUC model is 32.

(46) assumed to be isotropic with respect to the average geometric properties of the granular porous medium. Consequently, an isotropic RUC model is introduced by taking the average of one fully staggered array and two regular arrays, i.e. h √ i 1 × 12 2 + (1/ 2) + 2 × [12 (2 + 0)] F00 = × 3    . (1 − ǫ)4/3. 1 − (1 − ǫ). 1/3. . 1 − (1 − ǫ). .  2/3 2. .. (4.36). The existing RUC model also adopted the assumption of average geometrical isotropy, but the assumption was never mathematically justified as above. The final present RUC model for steady laminar flow of an incompressible Newtonian fluid through isotropic homogeneous granular porous media of low to moderate porosity in the limit of low Reynolds number flow, expressed as F00 , f00 d2 and K, are presented in Table 4.2. Dimensionless parameter. Isotropic RUC model. F00. 26.8 (1 − ǫ)4/3. f00 d2. . 1 − (1 − ǫ)1/3. . 1 − (1 − ǫ)2/3. 2. . 1 − (1 − ǫ)1/3. . 1 − (1 − ǫ)2/3. 2. . K. 26.8 (1 − ǫ)2/3. 1 − (1 − ǫ)1/3. . 1 − (1 − ǫ)2/3. 26.8 (1 − ǫ)4/3. 2. Table 4.2: Dimensionless expressions for the RUC model in the limit of low Reynolds number flow.. 4.2 4.2.1. Comparison with granular models from literature Hydraulic diameter. The hydraulic diameter is a length scale most commonly used for direct comparison between flow through systems of different geometrical structure. The hydraulic diameter Dh for capillary tube flow of uniform cross-section is defined as Dh = 4Rh ,. (4.37). where Rh denotes the hydraulic radius, defined as Rh =. cross-sectional area available for flow . wetted perimeter 33. (4.38).

(47) For flow through porous media Rh may also be expressed as (Bird et al. (2002)) Rh =. void volume/volume of bed ǫ = , wetted surface/volume of bed a. (4.39). where a is referred to as the specific surface and relates to the total solid surface per volume of particles av , i.e. the solid specific surface, through a = av (1 − ǫ) .. (4.40). The solid specific surface av is used to extend the definition of the hydraulic radius to account for beds of non-uniformly sized particles in order to obtain an average diameter. In effect, the assemblage of non-uniformly sized particles is replaced with an assemblage of uniformly sized particles, having the same ratio of total solid surface per volume of particles as the original assemblage, but not the same number of particles. Therefore the actual particle diameter does not enter into the determination of the hydraulic radius. The (average) hydraulic diameter may consequently be expressed in terms of the specific surface as follows 6 . (4.41) Dh = av The reason for the latter definition for the hydraulic diameter is to obtain the desired result of (4/3) πR3 6 Dh = = 6 = 2 R = Dp , (4.42) av 4 πR2 for an assemblage of uniformly sized spherical particles of radius R and diameter Dp . In RUC notation it follows that for an assemblage of uniformly sized cubes of length ds , Dh =. d3 6 = 6 s2 = ds . av 6 ds. (4.43). From equations (4.39) to (4.41) it follows that the relationship between the hydraulic radius and the hydraulic diameter may be expressed in terms of the porosity as Rh =. ǫ Dh . 6 (1 − ǫ). (4.44). The present RUC model is to be compared with other granular models from the literature. Based on the results of equations (4.42) and (4.43), it will henceforth be assumed that the diameter of a sphere in an assemblage of uniformly sized spherical particles is equal to the length of the cube ds in the RUC model.. 34.

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